Factoring Humanity

12

It was one of those moments of hazy semiconsciousness. Heather was dreaming—and knew that she was dreaming. And, like a good Jungian, she was trying to interpret the dream as it went along.

There was a cross in the dream. That in itself was unusual; Heather wasn’t given to religious symbolism.

But it wasn’t a wooden cross; rather, it was made of crystal. And it wasn’t a practical rendition—you couldn’t actually crucify a man on it. The arms were much, much thicker than they needed to be, and were rather stubby.

As she watched, the crystal cross began to rotate around its long axis. But as soon as it did so, it became apparent that it wasn’t really a cross. In addition to the protrusion at either side, there were identical protrusions front and back.

Her perspective was moving closer. She could see seams now; the object was made up of eight transparent cubes: a stack of them four high, and then four more arranged around the faces of the third cube from the top. It spun faster and faster, light glinting off its glassy surface.

An unfolded hypercube.

And, as she came even closer, she heard a voice.

Deep, masculine, resonant.

A strong voice.

The voice of God?

No, no—a superior being, but not God.

Her pattern suggests three-dimensional thinking.

Heather woke up, covered with sweat.

Spock, of course, had said his pattern in the film, referring to Khan. The “her”—well it had to be Heather, didn’t it?

Khan had been missing something—missing the obvious. Missing the fact that spaceships could go up and down as well as left and right or forward and backward. Heather had been missing something obvious, too, apparently—and her subconscious was trying to tell her that.

But as she lay there in bed, alone, she couldn’t figure out what.

“Good morning, Cheetah.”

“Good morning, Dr. Graves. You didn’t put me in suspend mode when you left yesterday; I took advantage of the time to do some online research, and I have some questions for you.”

Kyle headed over to the coffeemaker and set it about its business, then sat down in front of Cheetah’s console. “Oh?”

“I’ve been going through old news stories. I find that most electronic versions of newspapers only go back to some date in the nineteen-eighties or nineties.”

“Why should you care about decades-old news? It ain’t news if it’s old.”

“That was intended as a humorous comment, wasn’t it, Dr. Graves?”

Kyle grunted. “Yes.”

“I could tell by your use of the word ‘ain’t.’ You only use it when you’re trying to be funny.”

“Trust me, Cheetah, if you were human, you’d be rolling in the aisle.”

“And when you speak in a high tone like that, I know you’re still being funny.”

“Full marks. But you still haven’t told me why you’re reading old news stories.”

“You consider me to be non-human because, among other things, I can’t make ethical judgments that correspond to those a human would make. I have been looking for news stories that relate to ethical issues and am trying to fathom what a real human would do under such circumstances.”

“Okay,” said Kyle. “What story did you dig up that’s got you perplexed?”

“This: in nineteen eighty-five, a nineteen-year-old woman named Kathy was in her first year at Cornell University. On December twenty of that year, she was driving her boyfriend to his job at a grocery store in Ithaca, New York. The car hit a patch of ice, skidded ten meters, and slammed into a tree. The young man broke some bones, but a tire lying on the rear passenger seat pitched forward and hit Kathy’s head. She fell into a chronic vegetative state—essentially a coma—and was placed in the Westfall Healthcare Center in Brighton, New York. A decade later, in January, nineteen ninety-six, with her still in the coma, it was discovered that Kathy was pregnant.”

“How could she possibly be pregnant?” said Kyle.

“And that is the tone you use when speaking to me of matters of sexuality. You think that because I am a simulation that I could have no sophistication in such areas. But it’s you who are being naïve, Dr. Graves. The young woman was pregnant—indeed, had been pregnant for five months at the time it was discovered—because she had been raped.”

Kyle slumped slightly in his chair. “Oh.”

“The police launched a search for the rapist,” said Cheetah. “They came up with a list of seventy-five men who had had access to Kathy’s room, but the search quickly narrowed to a fifty-two-year-old certified nurse’s aide named John L. Horace. Horace had been fired three months previously for fondling a forty-nine-year-old multiple-sclerosis patient at Westfall. He refused to provide a DNA sample in the rape case, but police got some from an envelope flap and a stamp he had licked, and they determined that the odds were more than a hundred million to one in favor of Horace being the father.”

“I’m glad they caught him.”

“Indeed. In passing, though, I do wonder why this rapist gets automatic membership in the human race, but I have to prove myself?”

Kyle shuffled over to the coffeemaker, poured himself a cup. “That’s a very good question,” he said at last.

Cheetah was quiet for a time, then: “There’s more to this story.”

Kyle took a sip of coffee. “Yes?”

“There was the matter of the incidental zygotic commencement.”

“Ah, the coveted IZC. Oh, wait—you mean the baby. Christ, yes. What happened?”

“Prior to her accident, Kathy had been a devout Roman Catholic. She was, therefore, opposed to abortion. Taking that into account, Kathy’s parents decided that Kathy should have the baby and that they would raise the child.”

Kyle was incredulous. “Have the baby while still in a coma?”

“Yes. It is possible. Comatose women had given birth before, but this was the first known case of a woman becoming pregnant after going into a coma.”

“They should have aborted the pregnancy,” said Kyle.

“You humans make judgments so quickly,” said Cheetah, with what sounded like envy. “I have tried and tried to resolve this issue and I find I cannot.”

“What way are you leaning?”

“I tend to think that if they let the baby live, it should have been placed in a foster home.”

Kyle blinked. “Why?”

“Because Kathy’s mother and father, by forcing her to give birth in such extreme conditions, demonstrated that they were ill-suited to be parents.”

“Interesting take. Were there any polls conducted at the time about what should be done?”

“Yes—The Rochester Democrat Chronicle ran one. But the option I proposed wasn’t even put forward—meaning, I guess, that it’s not something a normal human would come up with.”

“No, it’s not. Your position has a certain logic to it, but it doesn’t seem right emotionally.”

“You said you would abort the child,” said Cheetah. “Why?”

“Well, I’m pro-choice, but even most of those who are pro-life make exceptions for cases of incest or rape. And what about the kid, for Pete’s sake? What effect would that kind of origin have on it?”

“That had not occurred to me,” said Cheetah. “The child—a boy—was born on March eighteen, nineteen ninety-six, and if he’s still alive, would be twenty-one now. Of course, his identity has been protected.”

Kyle said nothing.

“Kathy,” continued Cheetah, “died at the age of thirty, one day before the child’s first birthday; she never came out of the coma.” The computer paused. “It does make me wonder. The ethical dilemma—whether or not to countenance an abortion—could not have been drawn in sharper terms, even though I don’t seem to be able to resolve it properly.”

Kyle nodded. “We’re all tested in various ways,” he said.

“I know that better than most,” said Cheetah, in a tone that was a credible imitation of being rueful. “But when I am tested, it is by you. When human beings are tested, though—and a case such as this clearly seems to be a test—who is it that is administering the test?”

Kyle opened his mouth to reply, closed it, then opened it again. “That’s another very good question, Cheetah.”

Heather sat in her office, thinking.

She’d stared at the messages from space day in and day out for years, trying to fathom their meaning.

They had to be rectangular images. She’d tried to identify any cultural bias related to prime numbers, any reason why she’d interpret them one way while someone from China or Chad or Chile would interpret them some other way. But there wasn’t anything—the only cultural issue she could come up with was an argument about whether the number 1 qualified as a prime number.

No, if the length of the signals were the products of two prime numbers, then the only logical conclusion was that they were meant to be arranged into rectangular grids.

Her computer had all 2,843 messages stored on it.

But there were some messages that had been decoded, right at the beginning. Eleven of them, to be exact—a prime number. Meaning there were 2,832 undecoded messages.

Now that number was not a prime—it was an even number, and except for 2 there were, by definition, no even prime numbers.

A quantum computer could tell her in a twinkling what the factors of 2,832 were. Obviously, half that value would be a factor—1,416 would go into it twice. And half of that, 708. And half of that, 354. And half of that, 177. But 177 was an odd number, meaning that its half wouldn’t be a whole number.

She’d sometimes thought that maybe each day’s message made up only a portion of a larger whole, but she’d never found a meaningful way to order the pages. Of course, until a few days ago, they’d never known how many pages there were in total.

But now they did know. Maybe they did fit together into bigger groups, the way the backsides of trading cards often tile together to form a picture.

She brought up her spreadsheet program on her desktop computer and made a little sheet that simply divided 2,832 by consecutive integers, starting with 1.

There were only twenty numbers that divided into 2,832 evenly. She deleted the ones that didn’t divide evenly, leaving her with this table:

Of course, the assumption by most researchers was that there were 2,832 individual pages of data—but there might be as few as one page, made up of 2,832 tiles. Or there could be two pages, each made up of 1,416 tiles. Or three, made up of 944 tiles. And so on.

But how to tell which combination the Centaurs had intended?

She stared at the list, noting its symmetry: the first line was 1 and 2,832; the last was the reverse—2,832 and 1. And so the lines were paired up and down until the middle two: 48 and 59; 59 and 48.

It was almost as if the middle two were the pivot, the axle on which the great propeller of figures rotated.

And—

Christ—

Except for 1, 3, and 177, the number 59 was the only possibly prime number on the list: all the others were even numbers and, by definition, couldn’t be primes.

And—wait. Kyle had taught her a trick years ago. If the digits composing a number added up to a number divisible by three, then the original number was also divisible by three. Well, the digits making up 177—one, seven, and seven—added up to fifteen, and three went into fifteen five times, meaning 177 couldn’t be prime.

But what about the number 59? Heather had no idea how to determine if a number was prime, except by brute force. She made another quickie spreadsheet, this one dividing 59 by every whole number smaller than itself.

But none of them divided evenly.

None, except 1 and 59.

Fifty-nine was a prime number.

And—a thought occurred to her. One itself was sometimes considered a prime. Two was definitely a prime. So was three. But in a way, all those numbers were trivial primes: every whole number lower than them was also divisible only by itself or one. In many ways, five was the first interesting prime number—it was the first one in sequence that had numbers lower than itself that weren’t primes.

So if you discounted one, two, and three as trivially prime, then in the table she’d produced, 59 was the only non-trivial prime that divided evenly into the total number of undecoded alien messages.

It was another arrow pointing at that figure. The alien transmissions could possibly be arrayed in 48 pages each consisting of 59 individual messages, or 59 pages each consisting of 48 messages.

Researchers had been looking for recurring patterns in the messages for years, but so far none had turned up that hadn’t seemed coincidental. Now, though, that they knew the total number of messages, all sorts of fresh analyses could be done.

She opened another window on her computer and brought up the file directory of alien messages. She copied the directory into a text file, where she could play with it. She highlighted the bit counts for the first 48 undecoded messages and tallied them up: they totaled 2,245,124 bits. She then highlighted the next twenty-four. The tally came to 1,999,642.

Damn.

She then highlighted the counts for messages 12 through 71—the first 59 undeciphered messages.

The total came to 11,543,124 bits.

Then she highlighted messages 72 to 131 and tallied their sizes.

The total was also 11,543,124 bits.

Heather felt her heart pounding; perhaps someone had noticed this before, but. . .

She did it again, working her way through the material.

Her spirits fell when she found the fourth group tallied only 11,002,997 bits. But after a moment, she realized she’d highlighted only 58 messages instead of 59. She tried again.

The tally was 11,543,124.

She continued on until she’d done all 48 groupings of 59 messages.

Each group totaled precisely 11,543,124 bits.

She let out a great whoop! of excitement. Fortunately, her office did have that sturdy oak door.

The aliens hadn’t sent 2,832 separate messages—rather, they’d sent 48 large ones.

Now, if only she could figure out how to tile the messages together. Unfortunately, they were of many different sizes, and there was no orderly repetition from page to page. The first message making up the first group of 48 was 118,301 bits long (the product of the primes 281 and 421), whereas the first message of page two was 174,269 bits long (the product of the primes 229 and 761).

Presumably, the individual tiles formed square or rectangular shapes when properly placed together. She doubted she could figure it out by trial and error.

But surely Kyle could write her a computer program that would do it.

After last night, she was hesitant. What would she say to him?

She steeled her courage and picked up her phone.

“Hello?” said Kyle’s voice.

He doubtless knew it was Heather calling; he could read it off the status line on his phone. But there was no particular warmth in his voice.

“Hi, Kyle,” said Heather. “I need your help.”

Frosty: “You didn’t need my help last night.”

Heather sighed. “I’m sorry about that. Really I am. This is a difficult time for all of us.”

Kyle was silent. Heather felt the need to fill the void. “It’s going to take time to sort all this out.”

“I’ve been gone for a year now,” said Kyle. “How much time do you need?”

“I don’t know. Look, I’m sorry I called; I didn’t mean to disturb you.”

“That’s all right,” said Kyle. “Was there something?”

Heather swallowed, then went on. “Yes. I’ve had a breakthrough, I think, with the Centauri transmissions. If you take them in groups of fifty-nine messages, each group is exactly the same size.”

“Really?”

“Yes.”

“How many groups are there?”

“Exactly forty-eight.”

“So you think—what?—you think the individual messages form forty-eight bigger pages?”

“Exactly. But the individual pieces are all different sizes. I assume they fit together into a rectangular grid of some sort, but I don’t know how to work that out.”

Kyle made a noise that sounded like a snort.

“There’s no need to be condescending,” said Heather.

“No—no, that’s not it. Sorry. It’s just funny. See, this is a tiling problem.”

“Yes?”

“Well, this tiling problem—seeing if a finite number of tiles can be arrayed into a rectangular grid—is eminently solvable, just by brute-force computing. But there are other tiling problems that involve determining if specific tile shapes can cover an infinite plane, without leaving gaps that we’ve known since the nineteen-eighties fundamentally can’t be solved by a computer; if they’re solvable at all, it’s with an intuition that’s non-computable.”

“So?”

“So it’s just funny that the Centaurs would choose a message format that echoes one of the big debates in human consciousness, that’s all.”

“Hmm. But you say this is solvable?”

“Sure. I’ll need the dimensions of each message—the length and width in bits or pixels. I can write a program easily enough that will try sliding them around until they all fit together in a rectangular shape—assuming, of course, that there is such a pattern.” He paused. “There’ll be an interesting side effect, you know: if the individual tiles are not square and they all fit together only one way, you’ll know the orientation of each individual message. You won’t have to worry anymore about there being two possible orientations for each one.”

“I hadn’t thought of that, but you’re right. When can you do this?”

“Well, actually I’m too busy—sorry, but I am. But I can put one of my grad students on it. We should have an answer for you in a couple of days.

Heather tried to sound warm. “Thank you, Kyle.”

She could almost hear him shrug. “I’m always here for you, ” he said and clicked off.